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1. 1. Self-Adaptive Simulated Binary Crossover for Real-Parameter Optimization Kalyanmoy Deb*, S Karthik*, Tatsuya Okabe* GECCO 2007 *Indian Inst. India **Honda Research Inst., Japan Reviewed by Paskorn Champrasert paskorn@cs.umb.edu http://dssg.cs.umb.edu
2. 2. Outline • Simulated Binary Crossover (SBX Crossover ) (revisit) • Problem in SBX • Self-Adaptive SBX • Simulation Results – Self-Adaptive SBX for Single-OOP – Self-Adaptive SBX for Multi-OOP • Conclusion November 7, 08 DSSG Group Meeting 2/31
3. 3. SBX Simulated Binary Crossover • Simulated Binary Crossover (SBX) was proposed in 1995 by Deb and Agrawal. • SBX was designed with respect to the one-point crossover properties in binary-coded GA. – Average Property: The average of the decoded parameter values is the same before and after the crossover operation. – Spread Factor Property: The probability of occurrence of spread factor β ≈ 1 is more likely than any other β value. November 7, 08 DSSG Group Meeting 3/31
4. 4. Important Property of One-Point Crossover Spread factor: Spread factor β is defined as the ratio of the spread of offspring points to that of the parent points: c1 −c2 β= | p1 −p2 | c1 c2 - Contracting Crossover β<1 p1 p2 The offspring points are enclosed by the parent points. - Expanding Crossover β>1 c1 c2 The offspring points enclose by the parent points. p1 p2 - Stationary Crossover β=1 c1 c2 The offspring points are the same as parent points p1 p2 November 7, 08 DSSG Group Meeting 4/31
5. 5. Spread Factor (β) in SBX • The probability distribution of β in SBX should be similar (e.g. same shape) to the probability distribution of β in Binary-coded crossover. c(β) = 0.5(nc + 1)β nc , β ≤ 1 • The probability distribution function is defined as: c(β) = 0.5(nc + 1) β n1+2 , β > 1 c 1.6 1.4 Probability Distribution 1.2 1.0 n=2 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 5 6 7 β November 7, 08 DSSG Group Meeting 5/31
6. 6. A large value of n gives a higher probability for creating ‘near-parent’ contracting solutions. c(β) = 0.5(n + 1)β n , β ≤ 1 expanding 1 c(β) = 0.5(n + 1) β n+2 , β > 1 Mostly, n=2 to 5 November 7, 08 DSSG Group Meeting 6/31
7. 7. Spread Factor (β) as a random number • β is a random number that follows the proposed probability distribution function: 1.6 1.4 To get a β value, Probability Distribution 1.2 1) A unified random number u n=2 is generated [0,1] 1.0 0.8 0.6 2) Get β value that makes the 0.4 0.2 area under the curve = u 0.0 u 0 1 2 3 4 5 6 7 β Offspring: β c1 = x − 1 β(p2 − p1 ) ¯ 2 c2 = x + 1 β(p2 − p1 ) ¯ 2 November 7, 08 DSSG Group Meeting 7/31
8. 8. SBX Summary • Simulated Binary Crossover (SBX) is a real-parameter combination operator which is commonly used in evolutionary algorithms (EA) to optimization problems. – SBX operator uses two parents and creates two offspring. – SBX operator uses a parameter called the distribution index (nc) which is kept fixed (i.e., 2 or 5) throughout a simulation run. • If nc is large the offspring solutions are close to the parent solutions • If nc is small the offspring solutions are away from the parent solutions – The distribution index (nc) has a direct effect in controlling the spread of offspring solutions. November 7, 08 DSSG Group Meeting 8/31
9. 9. Research Issues • Finding the distribution index (nc) to an appropriate value is an important task. – The distribution index (nc) has effect in 1) Convergence speed if nc is very high the offspring solutions are very close to the parent solutions the convergence speed is very low. 2) Local/Global optimum solutions. Past studies of SBX crossover GA found that it cannot work well in complicated problems (e.g., Restrigin’s function) November 7, 08 DSSG Group Meeting 9/31
10. 10. Self-Adaptive SBX • This paper proposed a procedure for updating the distribution index (nc) adaptively to solve optimization problems. – The distribution index (nc) is updated every generation. – How the distribution index of the next generation (nc) is updated (increased or decreased) depends on how the offspring outperform (has better fitness value) than its two parents. • The next slides will show the process to make an equation to update the distribution index. November 7, 08 DSSG Group Meeting 10/31
11. 11. Probability density per offspring Spread factor distribution 1.6 1.4 (1) c(β) = 0.5(nc + 1)β nc , β ≤ 1 Probability Distribution 1.2 (1) (2) 1.0 nc = 2 (2) c(β) = 0.5(nc + 1) β n1+2 , β > 1 c 0.8 0.6 ¯ An example, when p1=2, p2=5 β>1 0.4 0.2 0.0 u 0 1 2 3 4 5 6 7 β β (1) (2) Offspring: ¯ 2¯ c1 = x − 1 β(p2 − p1 ) (1) (2) ¯ 2¯ c2 = x + 1 β(p2 − p1 ) November 7, 08 DSSG Group Meeting 11/31
12. 12. Self-Adaptive SBX (1) Consider the case that β > 1: c1 c2 β>1 p1 p2 c2 −c1 β= | p2 −p1 | (c2 −p2 )+(p2 −p1 )+(p1 −c1 ) β= p2 −p1 (c2 − p2 ) = (p1 − c1 ) 2(c2 −p2 ) β =1+ p2 −p1 --- (1) November 7, 08 DSSG Group Meeting 12/31
13. 13. Self-Adaptive SBX (2) 1.6 1.4 (1) c(β) = 0.5(nc + 1)β nc , β ≤ 1 Probability Distribution 1.2 (1) (2) 1.0 (2) c(β) = 0.5(nc + 1) β n1+2 , β > 1 c 0.8 0.6 0.4 0.2 0.0 u 0 1 2 3 4 5 6 7 β β (2) Rβ 0.5(nc +1) = (u − 0.5) --- (2) 1 β nc +2 dβ November 7, 08 DSSG Group Meeting 13/31
14. 14. (2) Self-Adaptive SBX (3) Rβ 0.5(nc +1) --- (2) 1 β nc +2 dβ = (u − 0.5) Rβ 1 0.5(nc + 1)β −(nc +2) dβ = (u − 0.5) −0.5β −nc −1 − (−0.5) = (u − 0.5) −0.5β −nc −1 = (u − 1) log(β −nc −1 ) = log − (u−1) 0.5 (−nc − 1) log β = log(−2(u − 1)) log 2(1−u) (−nc − 1) = log β log 2(1−u) nc = −(1 + log β ) --- (3) November 7, 08 DSSG Group Meeting 14/31
15. 15. Self-Adaptive SBX (4) c1 c2 c2 ˆ β>1 p1 p2 • If c2 is better than both parent solutions, – the authors assume that the region in which the child solutions is created is better than the region in which the parents solutions are. – the authors intend to extend the child solution further away from the closet parent solution (p2) by a factor α (α >1) (c2 − p2 ) = α(c2 − p2 ) ˆ --- (4) November 7, 08 DSSG Group Meeting 15/31
16. 16. β Update c1 c2 c2 ˆ β>1 p1 p2 2(c2 −p2 ) β =1+ p2 −p1 --- (1) (c2 − p2) = α(c2 − p2) --- (4) ˆ from (1) ˆ β =1+ 2(c2 −p2 ) ˆ p2 −p1 2((α(c2 −p2 )+p2 )−p2 ) =1+ p2 −p1 α2(c2 −p2 ) =1+ p2 −p1 β−1 ˆ β = 1 + α(β − 1) --- (5) November 7, 08 DSSG Group Meeting 16/31
17. 17. Distribution Index (nc) Update nc = −(1 + log 2(1−u) ) --- (3) ˆ β = 1 +α(β −1) --- (5) log β log 2(1−u) nc = −(1 + ˆ ˆ log β ) log 2(1−u) nc = −(1 + ˆ log(1+α(β−1)) ) log 2(1 − u) = −(nc + 1) log β (nc +1) log β nc = −1 + ˆ log(1+α(β−1)) --- (6) November 7, 08 DSSG Group Meeting 17/31
18. 18. Distribution Index (nc) Update When c is better than p1 and p2 then the distribution index for the next generation is calculated as (6). The distribution index will be decreased in order to create an offspring (c’) further away from its parent(p2) than the offspring (c) in the previous generation. (nc +1) log β nc = −1 + ˆ log(1+α(β−1)) --- (6) [0,50] α is a constant (α >1 ) November 7, 08 DSSG Group Meeting 18/31
19. 19. Distribution Index (nc) Update When c is worse than p1 and p2 then the distribution index for the next generation is calculated as (7). The distribution index will be increased in order to create an offspring (c’) closer to its parent(p2) than the offspring (c) in the previous generation. The 1/α is used instead of α (nc +1) log β nc = −1 + ˆ log(1+α(β−1)) --- (6) (nc +1) log β nc = −1 + ˆ log(1+(β−1)/α) --- (7) α is a constant (α >1 ) November 7, 08 DSSG Group Meeting 19/31
20. 20. Distribution Index (nc) Update So, we are done with the expanding case (2) How about contracting case (1) ? The same process is applied for (1) 1.6 1.4 (1) c(β) = 0.5(nc + 1)β nc , β ≤ 1 Probability Distribution 1.2 1.0 (2) c(β) = 0.5(nc + 1) β n1+2 , β > 1 c 0.8 (1) (2) 0.6 0.4 0.2 0.0 u 0 1 2 3 4 5 6 7 β β (1) Rβ 0 0.5(nc + 1)β nc dβ = u --- (x) November 7, 08 DSSG Group Meeting 20/31
21. 21. Distribution Index (nc) Update for contracting case When c is better than p1 and p2 then the distribution index for the next generation is calculated as (8). The distribution index will be decreased in order to create an offspring (c’) father away from its parent(p2) than the offspring (c) in the previous generation. 1+nc --- (8) nc = ˆ α −1 When c is worse than p1 and p2, 1/α is used instead of α nc = α(1 + nc ) − 1 ˆ --- (9) α is a constant (α >1 ) November 7, 08 DSSG Group Meeting 21/31
22. 22. Distribution Index Update Summary Expanding Case Contracting Case (u > 0.5) (u<0.5) c is better than parents 1+nc nc = −1 + ˆ (nc +1) log β ˆ nc = α −1 log(1+α(β−1)) c is worse than parents (nc +1) log β nc = −1 + ˆ log(1+(β−1)/α) nc =α(1+nc) −1 ˆ (u = 0.5), nc = nc ˆ November 7, 08 DSSG Group Meeting 22/31
23. 23. Self-SBX Main Loop Update nc If rand < pc Parent 1 Offspring C1 Pt Selection Crossover initial Parent 2 Offspring C2 population If rand > pc N = 150 Mutation nc = 2 α = 1.5 Pt+1 Population at t+1 N = 150 α = 1.5 November 7, 08 DSSG Group Meeting 23/31
24. 24. Simulation Results Sphere Problem: SBX • Initial nc is 2. • Population size =150 • Pc = 0.9 • The number of variables (n) =30 • Pm =1/30 • A run is terminated when a solution having a function value = 0.001 • 11 runs are applied. Self-SBX • Pc = 0.7 • α = 1.5 November 7, 08 DSSG Group Meeting 24/31
25. 25. Fitness Value • Self-SBX can find better solutions November 7, 08 DSSG Group Meeting 25/31
26. 26. α Study • α is varied in [1.05,2] • The effect of α is not significant since the average number of function evaluations is similar. November 7, 08 DSSG Group Meeting 26/31
27. 27. Simulation Results Rastrigin’s Function: Many local optima, one global minimum (0) The number of variables = 20 SBX • Initial nc is 2. • Population size =100 • Pc = 0.9 • The number of variables (n) =30 • Pm =1/30 • A run is terminated when a solution having a function value = 0.001 or the maximum of 40,000 generations is reached. • 11 runs are applied. Self-SBX • Pc = 0.7 • α = 1.5 November 7, 08 DSSG Group Meeting 27/31
28. 28. Fitness Value Best Median Worst • Self-SBX can find a very close global optimal November 7, 08 DSSG Group Meeting 28/31
29. 29. α Study • α is varied in [1.05,10] • The effect of α is not significant since the average number of function evaluations is similar. • α = 3 is the best November 7, 08 DSSG Group Meeting 29/31
30. 30. Self-SBX in MOOP • Self-SBX is applied in NSGA2-II. • Larger hyper-volume is better November 7, 08 DSSG Group Meeting 30/31
31. 31. Conclusion • Self-SBX is proposed in this paper. – The distribution index is adjust adaptively on the run time. • The simulation results show that Self-SBX can find the better solutions in single objective optimization problems and multi-objective optimization problems. • However, α is used to tune up the distribution index. November 7, 08 DSSG Group Meeting 31/31
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